The exact same thing has been happening to my 99 LX470. I talked to my local Toyota/Lexus shop and they told me first try the connections on the battery. Sometimes a film can develop on the terminals even though they look clean. I cleaned them up real good and actually scored them with a wire brush. I also did the insides of the connections on the wires that clamp to the terminals.
No problems since.
They also told me that if what I just descibed didn't fix the problem it would probably be the starter solenoids/connections.
Good luck!
